
From the expression of the coefficients of \(f(z)\)  it is clear that \(D<0\) . Now suppose \(A<0\) , then from  we can infer that the product of the roots is negative. In this case either we will have two positive root and one negative root or the three roots will be negative. Now using superradiance condition REF  we can see that \(C>0\) . Then from  it is clear that we can never have three negative roots if \(A<0\) .
In this case we are left with the possibility of having two positive roots and one negative root. That is, with the condition \(A<0\)  one can not avoid the possibility of forming trapping well outside the horizon. On the other hand, if we let \(A>0\) , then the product of the roots become positive. In this case, we can choose our parameters to have only one positive (say \(z_{1}\) ) and two negative roots (say \(z_{2}\)  and \(z_{3}\) ). So the necessary condition to avoid trapping well is \(A>0\) (see also [1]}). In terms of perturbation parameters this gives,
\(\omega ^{2}<\frac{\mu ^{2}}{2}~.\) 
